{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 模型选择、过拟合、欠拟合\n",
    "\n",
    "当模型在训练集上更准确时，它在测试数据集上却不一定更准确。\n",
    "\n",
    "- **训练数据集** 用于模型训练的数据集，对应的误差为**训练误差**。 \n",
    "- **测试数据集** 用于测试训练后的模型性能的数据集，通常使用测试集的误差来近似**泛化误差**。\n",
    "\n",
    "在模型选择的时候，就要选择使泛化误差最小的模型。\n",
    "\n",
    "但从严格意义上讲，**测试集只能在所有超参数和模型选择参数选定后使用一次，不能使用测试数据对模型进行调参。** \n",
    "\n",
    "由于，无法从训练误差估计泛化误差，因此也不能只能训练数据集选择模型。 因此，通常在训练数据集和测试数据集外的数据进行模型选择，这部分数据称为验证数据集。 通常来说，可从给定的训练数据集中随机选择一小部分作为验证集，而将剩余的部分作为真正的训练集。\n",
    "\n",
    "** K折交叉验证** \n",
    "把原始训练数据集随机的分割为$K$个不重合的子数据集，然后做$K$此模型训练和验证。　每一次，使用一个子数据集作为验证集，并使用其他的$K-1$子数据集来训练模型。　在这$K$次训练和验证中，每次用来验证模型的子数据集都不同。　最后，对这$K$次训练误差和验证误差分别求平均。\n",
    "\n",
    "### 过拟合、欠拟合\n",
    "\n",
    "给定的数据集，如果模型的复杂度过低，很容易出现欠拟合；模型的复杂度过高，很容易出现过拟合。　欠拟合，在训练时无法得到较低的训练误差；过拟合，则在测试集上误差远小于训练误差。　\n",
    "\n",
    "影响欠拟合和过拟合的一个重要因素就是训练数据集的大小。　一般来说，如果训练样本中数据过少，特别市比模型参数量更少时，容易发生过拟合。　另外，泛化误差不会随着训练样本的增大而增加，因此在允许的范围内，总是希望训练数据集大一些。　"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 多项式函数拟合实验\n",
    "\n",
    "生成一个人工数据集，给定样本特征$x$，使用如下三阶函数生成该样本的标签\n",
    "$$\n",
    "y = 1.2x - 3.4x^2 + 5.6x^3 + 5 + \\sigma\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 矩阵乘法的分解\n",
    "$$\n",
    " \\begin{aligned} XW &= \\left[\\begin{array}{c}x_{11} & x_{12} & x_{13} \\\\ x_{21} &x_{22} &x_{23} \\\\ x_{31} & x_{32} &x_{33}\\end{array}\\right] \\cdot \\left [\\begin{array}{c} w_{11} \\\\w_{21} \\\\ w_{31} \\end{array}\\right] \\\\\n",
    "&=\\left[\\begin{array}{c}x_{11} \\\\ x_{21} \\\\ x_{31} \\end{array} \\right] \\cdot [w_{11}] + \\left[\\begin{array}{c}x_{12} \\\\ x_{22} \\\\ x_{32} \\end{array} \\right] \\cdot [w_{21}] + \\left[\\begin{array}{c}x_{13} \\\\ x_{23} \\\\ x_{33} \\end{array} \\right] \\cdot [w_{31}]\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "tensor([[ 1.2856],\n        [-0.6947]]) tensor([[ 1.2856,  1.6528,  2.1248],\n        [-0.6947,  0.4826, -0.3353]]) tensor([24.0695,  3.9429])\n"
    }
   ],
   "source": [
    "import torch\n",
    "import numpy as np \n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "n_train,n_test,true_w,true_b = 100,100,[1.2,3.4,5.6],5\n",
    "\n",
    "# 生成数据集\n",
    "features = torch.randn((n_train+ n_test,1))\n",
    "poly_features = torch.cat((features,torch.pow(features,2),torch.pow(features,3)),1)\n",
    "\n",
    "labels = (true_w[0] * poly_features[:,0] + true_w[1] * poly_features[:,1] + true_w[2] * poly_features[:,2] + true_b)\n",
    "labels += torch.tensor(np.random.normal(0,0.01,size=labels.size()),dtype=torch.float32)\n",
    "\n",
    "print(features[:2],poly_features[:2],labels[:2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def semilogy(x_vals,y_vals,x_label,y_label,x2_vals=None,y2_vals=None,legend=None):\n",
    "    plt.xlabel(x_label)\n",
    "    plt.ylabel(y_label)\n",
    "    plt.semilogy(x_vals,y_vals)\n",
    "    if x2_vals and y2_vals:\n",
    "        plt.semilogy(x2_vals,y2_vals,linestyle=':')\n",
    "        plt.legend(legend)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_epochs = 100\n",
    "loss = torch.nn.MSELoss()\n",
    "\n",
    "def fit_and_plot(train_features,test_features,train_labels,test_labels):\n",
    "    net = torch.nn.Linear(train_features.shape[-1],1)\n",
    "    batch_size = min(10,train_labels.shape[0])\n",
    "    dataset = torch.utils.data.TensorDataset(train_features,train_labels)\n",
    "    train_iter = torch.utils.data.DataLoader(dataset,batch_size,shuffle=True)\n",
    "\n",
    "    optimizer = torch.optim.SGD(net.parameters(),lr=0.01)\n",
    "    train_ls = []\n",
    "    test_ls = []\n",
    "\n",
    "    for _ in range(num_epochs):\n",
    "        for X,y in train_iter:\n",
    "            l = loss(net(X),y.view(-1,1))\n",
    "            optimizer.zero_grad()\n",
    "            l.backward()\n",
    "            optimizer.step()\n",
    "\n",
    "        train_labels = train_labels.view(-1,1)\n",
    "        test_labels = test_labels.view(-1,1)\n",
    "        train_ls.append(loss(net(train_features),train_labels).item())\n",
    "        test_ls.append(loss(net(test_features),test_labels).item())\n",
    "    print('final epoch:train loss',train_ls[-1],'test loss',test_ls[-1])\n",
    "    semilogy(range(1,num_epochs+1),train_ls,'eopchs','loss',range(1,num_epochs+1),test_ls,['train','test'])\n",
    "    print('weight:',net.weight.data,'\\nbias:',net.bias.data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "final epoch:train loss 0.00010008071694755927 test loss 0.00012053922546328977\nweight: tensor([[1.2037, 3.4005, 5.5992]]) \nbias: tensor([5.0007])\n"
    },
    {
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fxn/mbtTBbaVUjdIns+uI4c4VDPH/jIsWn4PDx8FDg9ohInaHpZSqBzRR1BXn3Ilv+2H0/ymdqQs24ZtxgPuvOg+HQ5OFUsqztOuprvBx4ohsxlNXdGZi218Zt24Ez8+cg0vHLJRSHuZVicLbpseejogw+IqbWNHsdl5bZ3jyax2zUEp5llclCm+bHlsaiWxB/1ue5pZ+Lfj651VMn7PY7pCUUl7MqxJFfSIiPD6kLXNCn6LVsvG8s3Sn3SEppbyUDmbXYQ5fJ+EjXmbeokze/3wDDodwwznN7A5LKeVltEVRx/m2u4S/3zyMC9rHsvrz13hn4W92h6SU8jKaKLxAgNOHSRc5ecHvDbZ98zqTF26zOySllBfRricv4YzvTsHoeRz92ZfpczbRMCyA4d3j7A5LKeUFan2LQkRaishUEfnI7lhqO9/mfXjh+p5cHp9L8Kd/YtP2nXaHpJTyAh5NFCIyTUQOicj6EscHi8hmEdkqIo+UdQ9jzHZjzBhPxulNnD4O/n1JY3rIVv47aw5HM3XFWaVU1Xi6RTEdGFz8gIj4AK8CQ4COwEgR6SgiXUTkqxI/sR6Ozys1aNuHvTcvZ35WC+5+L4mc/EK7Q1JK1WEeTRTGmIXA0RKHewFb3S2FPGAWMNwYs84Yc3mJn0OejM+bdW3RiKev7ELHXe+w7T/ncuhoyf8NSilVPnaMUcQBe4qVU9zHTktEokTkDaCHiIwv47yxIpIkIkmHDx+uvmjrsBE94xnauxPJuQ24YtIa1u9Ng/xsu8NSStUxdiSK0y13WupiRcaYVGPMncaYVsaYZ8o4bzLwT2C1n59fNYTpHboOHUfrO94D4IE3PiX3xW6wea7NUSml6hI7EkUKkFCsHA/sq44b15e1niqqc1w4n43rR4OoGBZkNuOnY5F2h6SUqkPsSBQrgTYi0kJE/IDrgS+q48b1YfXYyooNDWDSHZcwqdET3PpFKp+uSYGcdLvDUkrVAeVKFCJyr4iEiWWqiKwWkUvKcd1MYCnQTkRSRGSMMaYAGAd8A2wEZhtjNlSlEkW0RVG28EAn74w5h3NaRHLo44fJeqU/FObbHZZSqpYrb4viVmNMOnAJEAPcAkw400XGmJHGmMbGGKcxJt4YM9V9fI4xpq173OGpSkdfgrYozizY35dpo89mV1hP3snux4nsHLtDUkrVcuVNFEUD0JcCbxljfuH0g9K20hZF+QQ4fRhx/S1MyLqc/3y/y+5wlFK1XHkTxSoR+RYrUXwjIqGAy3NhKU87q2kDRvdtztYVc9iy6EO7w1FK1WLlXRRwDNAd2G6MyRKRSKzup1pFRIYCQ1u3bm13KHXCXy9uy87Vo8mfL+T0vooAp4/dISmlaqHytij6AJuNMcdF5EbgMaDWDQRo11PFBAc4yRg6mauyxvOX2WvJL9RGolLqj8qbKF4HskSkG/AQsAt422NRqRpzzlk9ePCybny3LoX3pzxLQYGuC6WUOlV5E0WBMcYAw4GJxpiJQKjnwqocnfVUObcNaMnUbpu5+cAEJr49kwJtWSiliilvojjhXmfpJuBr9wqwTs+FVTna9VR55173Vz7rPoX/bWnAmBlJ5H4yDtbpFiBKqfIniuuAXKznKQ5gLeL3nMeiUjXP4eCKK67l6Su7sGzbIfasW8Sh3Zut94wBl3ZJKVVflStRuJPDe0C4iFwO5Bhjat0YhXY9Vd2oc5oy845+3OB8gXOXncWcdfth/cfw5kWQqntxK1UflXcJj2uBFcA1wLXAchEZ4cnAKkO7nqrHWU0b8NXdA+jUJJw/v7+aBdtPQHwi+Ne6YSmlVA0ob9fTo8DZxpibjTF/wtp86HHPhaXsFhPqz3u3ncOF7WMZvTSW/zpvxwTH2B2WUsoG5U0UjhK7zaVW4FpVRwU4fXjjxp5c0zOeiT8kM/XTuTDnIR2vUKqeKe+T2fNE5Btgprt8HTDHMyGp2sTXx8F/RnTF6etg9crZ5Ae9j7PnaGjY0e7QlFI1pLyD2Q8Ck4GuQDdgsjHmYU8GVhk6mO0ZIsITQzuRmjCYfrn/ZX1BqTvXKqW8kFjP0XmXxMREk5SUZHcYXufwiVyGvbIYhwhzhrkIb9YDgqPsDkspVU1EZJUxJrHk8TJbFCJyQkTST/NzQkR0e7R6JibUn0k39WRI5qeEz76a41sW2R2SUqoGlDlGYYzR+ZDqFF3jI0i94XGGv9+Fo3MDmdwonQ7pP0NsR2jQzO7wlFIeoDOXVIWd3z6Wp++8jjx8GPPGdxR8dDsset7usJRSHqKJQlVKpybhfP7n/kRExnJ11iN8Hf+A3SEppTzEqxKFznqqWY3CA5h1R2/8E3oy7sMNfLA0GQ6sszsspVQ186pEoUt41LywACczbu3FwLYx+M65n7wpgzBZR+0OSylVjbwqUSh7BPr5MOmmRFa1/Qt3Z9/Oo9/stfa08MKp10rVR5ooVLXw83Xw5A3n0+rckby/fDcTJk2nYPpQSN9nd2hKqSoq7xIeSp2RwyE8NLg9TSODWPT5Cnb6p2DSfWgTZndkSqmq0BaFqnbX92rKqFvuZqQ8y+VvrOa9pdsxs0ZB0lt2h6aUqgRNFMoj+rWO5uv7zqNXi0he+Xwx6/YcI9/hb73pckFmqr0BKqXKTROF8pjY0ABm3NKLmwb3ZVjqOO5a15a8Ahd8cTfMGAr52XaHqJQqhzoxRiEiVwCXAbHAq8aYb20OSZWTwyH838DWhPr78vjnG7h75mpe7X0Vvg07gW+A3eEppcrB4y0KEZkmIodEZH2J44NFZLOIbBWRR8q6hzHmM2PM7cBorL0wVB1zU5/m/GNoR77ZcJB7VkSQd/adIAIFuXaHppQ6g5poUUwHXgHeLjogIj7Aq8DFQAqwUkS+AHyAZ0pcf2ux3fUec1+n6qBb+rWg0GV48uuNZOQmMemSQAJnXwcXPA7dR9odnlKqFB5PFMaYhSLSvMThXsBWY8x2ABGZBQw3xjwDXF7yHiIiwARgrjFm9ek+R0TGAmMBmjZtWm3xq+p124CWhAU4eeSTXxmdFcDbTQfgn9DL7rCUUmWwazA7DthTrJziPlaau4GLgBEicufpTjDGTDbGJBpjEmNiYqovUlXtrj07gddv7MmaA3lcumskO0wj641vH4MVU6zXuSdg+0/2BamU+p1diUJOc6zU9R6MMS8bY3oaY+40xrxR6k11UcA6Y1CnRrxzay+OZuYx7JXFLFr9K6QkQWG+dcK88TBrFGQfszdQpZRtiSIFSChWjgeqvNaDLgpYt5zTMoov7+5Ps6ggbpq9hxfjJ1J4zl3Wm+c/Cjd+AoEN7A1SKWVbolgJtBGRFiLiB1wPfFHVm2qLou6JbxDER3f2ZUTPeF7+cSsj3ljC9sMZENYYmp5jnaStCqVsVRPTY2cCS4F2IpIiImOMMQXAOOAbYCMw2xizoaqfpS2KuinA6cNzI7oy8frubD+cyZCJi5i6eIe1Au3GL+G/XeBglf96KKUqSYwXLQUtIkOBoa1bt749OTnZ7nBUJRxKz2H8J+v4YdMhEiIDueecCK46/Do+wyaCX5Dd4Snl1URklTEmseRxr1rCQ1sUdV9sWABv3pzIm39KJDrEnwfn7qf35utI2pdjPZy3Z6XdISpV72iLQtVaxhiWbT/K3z5dx7GsPOZ3X0CDX6fCA5shMMLu8JTyOtqiUHWOiNCnVRQzbumFjwijNvbhxPlPnUwS+3+Bgjx7g1SqHvCqRKG8U9OoIKaOPpsdGb7cuKY9WXkFcHQ7TLkAkqZZJxUWwMo3IT/H3mCV8kJelSh0eqz36p4Qwf9GnsW6vWmMmrKcI76NYMQ0aHW+dcLW7+HrB2D7AlvjVMobedUYRZHExESTlJRkdxjKA77ZcIB7Z60hJtSft0b3onVsCAWFLnYdzSI+ayP+TROtVWmP74bwBOu1UqpcShujqBP7UShVZFCnRswa24fbZqzkqtd+JiEyiORDGeQVuLisa2NebSZwJBkmD4QLHoPed9kdslJ1nld1Pan6oXtCBJ/+Xz+6JUQQGezH6L7NuaZnPF//up/5mw5BVGvoMw46Drc7VKW8gld1Pen02Porr8DFpS8vIjuvkO/+ci5Bfu7Gcl4mLHkFBj4M+9bC3IfgmhnWEiFKqVPo9Fjl1fx8HTx9ZRf2Hs9m4vfFfkn46VlY+Bwc3wMB4da6Uel77QtUqTrIqxKFqt96tYjk+rMTeHPxDjbsc89863sPPLAJIhLYZWLZdf18iP/DL0xKqTJoolBe5ZEh7WkQ5GT0WytZtesYBEdDcDRz1+1n0EsLuX7KcvLz82DXErtDVarO8KpEoc9RqIggP96/vTdBfj6MnLyM2Sv38Or8rdz13mpiQv3Zn5bDlk+ehOmXQeo2u8NVqk7wqsHsIvochTqelce499eweOsRAIZ3b8KzV3dl6P8WEyknmDWoEOkwzDpZn7VQCqgng9lKFYkI8mP6LWdz74VteOyyDrx0XXcCnD7cPqAlyw8Ki519rQSx6i2YeskfN0dyuWDh89YguFL1nCYK5bV8fRzcf3FbbhvQEnG3Gob3aEJsqD+TF263TvIPg5CGEOBeaNDlsv5MT4GfJ8KvH9gQuVK1iyYKVa/4+/owul9zFiUf4bd96dBlBFz3jtW6yD4Gb14AhzZBRFO4czEMeMAay8jNsDt0pWyjiULVOzf0akaQnw8vfb/F2m61SO4JcDjhwDqr3KCZtUrtK4mweoY9wSpVC3hVotBZT6o8woOc3HVeK7797SDXTFrKrtRM642IpnDrN9B20MmTo1rBpc9D5xH2BKtULaCznlS99eUv+3j003UUugz/GNaJa3rG/z6WoVR9pLOelCphaLcmzLvvXLrEh/PQR78yZkYSB9NL2fjowDr46v7Sxyq88BcupYpoolD1WpOIQN6/rTd/v7wjS7Yd4eIXf+LdZbs4kpF76onHdsGvH0J+llUuyIOCXGuW1Jf3wfynaz54pWqI7keh6j2HQ7i1fwvObx/LQx/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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "fit_and_plot(poly_features[:n_train,:],poly_features[n_train:,:],labels[:n_train],labels[n_train:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 线性函数拟合（欠拟合）\n",
    "\n",
    "使用线性函数来拟合非线性模型，会出现欠拟合。模型的训练误差在迭代的早期下降后便很难在继续降低，在最训练误差保持在一个较大的值，不再下降."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "final epoch:train loss 226.62135314941406 test loss 677.64404296875\nweight: tensor([[18.9664]]) \nbias: tensor([8.9288])\n"
    },
    {
     "output_type": "display_data",
     "data": {
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "fit_and_plot(features[:n_train,:],features[n_train:,:],labels[:n_train],labels[n_train:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 训练样本不足（过拟合）\n",
    "事实上，即便使用与数据生成模型同阶的三阶多项式函数，如果训练样本不足，该模型依然很容易过拟合。显然，训练样本过少了了，甚⾄至少于模型参数的数量量。这使模 型显得过于复杂，以⾄至于容易易被训练数据中的噪声影响。在迭代过程中，尽管训练误差较低，但是测试 数据集上的误差却很⾼高。这是典型的过拟合现象。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "final epoch:train loss 0.10633235424757004 test loss 9.739349365234375\nweight: tensor([[1.7816, 4.6718, 4.7165]]) \nbias: tensor([4.0462])\n"
    },
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "fit_and_plot(poly_features[0:2,:],poly_features[n_train:,:],labels[0:2],labels[n_train:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 总结\n",
    "- 由于⽆无法从训练误差估计泛化误差，一味地降低训练误差并不不意味着泛化误差⼀一定会降低。机器器学习模型应关注降低泛化误差。\n",
    "- 可以使⽤用验证数据集来进⾏行行模型选择。\n",
    "- ⽋欠拟合指模型⽆无法得到较低的训练误差，过拟合指模型的训练误差远⼩小于它在测试数据集上的误差\n",
    "- 应选择复杂度合适的模型并避免使⽤用过少的训练样本。"
   ]
  }
 ],
 "metadata": {
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0-final"
  },
  "orig_nbformat": 2,
  "kernelspec": {
   "name": "python3",
   "display_name": "Python 3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}